integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
The special case of fiber integration in differential cohomology for ordinary differential cohomology is the partial higher holonomy operation for circle n-bundles with connection:
for $Y \to X$ a bundle of compact smooth manifolds $S$ of dimension $k$ and $[\nabla] \in H_{diff}^n(Y)$ a class in ordinary differential cohomology of degree $n$ on $Y$, its fiber integration
is a differential cohomology class on $X$ of degree $k$ less.
In the particular case that $X = *$ is the point and $dim Y = k = n-1$ the element
is the higher holonomy of $\nabla$ over $Y$.
The operation of fiber integration in generalized (Eilenberg-Steenrod) cohomology requires a choice of orientation in generalized cohomology. For fiber integration in differential cohomology this is to be refined to a differential orientation .
Accordingly, instead of a Thom class there is a differential Thom class .
For $X$ a compact smooth manifold and $V \to X$ a smooth real vector bundle of rank $k$ a differential Thom cocycle on $V$ is
a compactly supported cocycle $\hat \omega$ in the ordinary differential cohomology of degree $k$ of $V$;
such that for each $x \in X$ we have
The underlying class $[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z})$ in compactly supported integral cohomology is an ordinary Thom class for $V$.
Let $p : X \to Y$ be a smooth function of smooth manifolds.
An $H \mathbb{Z}_{diff}$-orientation on $p$ is
A factorization through an embedding of smooth manifolds
for some $N \in \mathbb{N}$;
a tubular neighbourhood $W \hookrightarrow Y \times \mathbb{R}^N$ of $X$;
a differential Thom cocycle, def. 1, $U$ on $W \to X$.
This appears as (HopkinsSinger, def. 2.9).
Write $H^n_{diff}(-)$ for ordinary differential cohomology. For any choice of presentation, there is a fairly evident fiber integration of compactly supported cocycles along trivial Cartesian space bundles $Y \times \mathbb{R}^N \to Y$ over a compact $Y$:
Let $X \to Y$ be a smooth function equipped with differential $H\mathbb{Z}$-orientation $U$, def. 2. Then the corresponding fiber integration of ordinary differential cohomology is the composite
This appears as (HopkinsSinger, def. 3.11).
We discuss an explicit formula for fiber integration along product-bundles with compact fibers in terms of Deligne complex, following (Gomi-Terashima).
For $X$ a smooth manifold, write $\mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$ for the Deligne complex in degree $(n+1)$ over $X$.
Let $X$ be a paracompact smooth manifold and let $F$ be a compact smooth manifold of dimension $k$ without boundary. Then there is a morphism
given by (…)
(Gomi-Terashima, section 2, corollary 3.2)
The above formulation of fiber integration in ordinary differential cohomology serves as a presentation for a more abstract construction in smooth homotopy theory.
Let $\mathbf{H} \coloneqq$ Smooth∞Grpd be the ambient cohesive (∞,1)-topos of smooth ∞-groupoids/smooth ∞-stacks. As discussed there, the Deligne complex, being a sheaf of chain complexes of abelian groups, presents under the Dold-Kan correspondence a simplicial presheaf on the site CartSp, which in turn presents an object
discussed here: the smooth moduli ∞-stack of circle n-bundles with connection.
Let now $\Sigma_k$ be a compact smooth manifold of dimension $k \in \mathbb{N}$ without boundary. There is the internal hom in an (infinity,1)-topos
which is the smooth moduli $n$-stack of circle $n$-connections on $\Sigma_k$.
For all $k \leq n$ there is a natural morphism
which for $U \in$ SmthMfd a smooth test manifold sends $n$-connections on $\Sigma_k$ on $U \times \Sigma_k$ to the $(n-k)$-connection on $U$ which is their fiber integration over $\Sigma_k$.
To see this, observe that
by definition $\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn})$;
if $\{U_i \to \Sigma_k\}$ is a fixed good open cover of $\Sigma_k$, then $\{U \times U_i \to U \times \Sigma_k\}$ is also a good open cover, for every $U \in$ CartSp;
hence the Cech nerve $C(\{U \times U_i\})$ is a natural (functorial in $U \in CartSp$) cofibrant object resolution of $U \times \Sigma_k$ in the projective local model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,loc}$ which presents $\mathbf{H} =$Smooth∞Grpd (as discussed there);
the (image under the Dold-Kan correspondence) of the Deligne complex $\mathbb{Z}(n+1)^\infty_D$ is a is fibrant in this model structure (since every circle $n$-bundle is trivializable over a contractible space $U \in$ CartSp).
This means that a presentation of $[\Sigma_k, \mathbf{B}^n U(1)_{conn}]$ by an object of $[CartSp^{op}, sSet]_{proj,loc}$ is given by the simplicial presheaf
that sends $U$ to the Cech-Deligne hypercohomology chain complex with respect to the cover $\{U \times U_i \to U \times \Sigma_k\}$.
On this def. 4 provides a morphism of simplicial sets
which one directly sees is natural in $U$, hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in $\mathbf{H}$.
Applications are to
(…)
At least the fiber integration all the way to the point exists on general grounds for the intrinsic differential cohomology in any cohesive (∞,1)-topos: the general abstract formulation is in the section Higher holonomy and Chern-Simons functional and the implementation in smooth ∞-groupoids is in the section Smooth higher holonomy and Chern-Simons functional .
(…)
Differential universal characteristic class / extended $\infty$-Chern-Simons Lagrangian:
moduli $\infty$-stack of higher gauge fields on a given $\Sigma_k$:
Lagrangian of $\hat \mathbf{c}$-Chern-Simons theory:
extended action functional of $\hat \mathbf{c}$-Chern-Simons theory in codimension $(n-k)$
(…)
A discussion in the general sense of fiber integration in generalized (Eilenberg-Steenrod) cohomology is in section 3.4 of
and around prop. 2.1 (in the context of Chern-Simons theory) in
Explicit formulas for fiber integration of cocycles in Cech-Deligne cohomology are given in
and their generalization from higher holonomy to higher parallel transport in
and
See also
The observation that the construction in Gomi-Terashima induces refines to smooth higher moduli stacks is discussed in
for the case without boundary and for the general case in